Theorem cofunexg 5680

Description: Existence of a composition when the first member is a function. (Contributed by NM, 8-Oct-2007.)

Assertion

Ref	Expression
cofunexg	⊢ ((Fun A ∧ B ∈ 𝐶) → (A ∘ B) ∈ V)

Proof of Theorem cofunexg

Step	Hyp	Ref	Expression
1		relco 4762	. . 3 ⊢ Rel (A ∘ B)
2		relssdmrn 4784	. . 3 ⊢ (Rel (A ∘ B) → (A ∘ B) ⊆ (dom (A ∘ B) × ran (A ∘ B)))
3	1, 2	ax-mp 7	. 2 ⊢ (A ∘ B) ⊆ (dom (A ∘ B) × ran (A ∘ B))
4		dmcoss 4544	. . . . 5 ⊢ dom (A ∘ B) ⊆ dom B
5		dmexg 4539	. . . . 5 ⊢ (B ∈ 𝐶 → dom B ∈ V)
6		ssexg 3887	. . . . 5 ⊢ ((dom (A ∘ B) ⊆ dom B ∧ dom B ∈ V) → dom (A ∘ B) ∈ V)
7	4, 5, 6	sylancr 393	. . . 4 ⊢ (B ∈ 𝐶 → dom (A ∘ B) ∈ V)
8	7	adantl 262	. . 3 ⊢ ((Fun A ∧ B ∈ 𝐶) → dom (A ∘ B) ∈ V)
9		rnco 4770	. . . 4 ⊢ ran (A ∘ B) = ran (A ↾ ran B)
10		rnexg 4540	. . . . . 6 ⊢ (B ∈ 𝐶 → ran B ∈ V)
11		resfunexg 5325	. . . . . 6 ⊢ ((Fun A ∧ ran B ∈ V) → (A ↾ ran B) ∈ V)
12	10, 11	sylan2 270	. . . . 5 ⊢ ((Fun A ∧ B ∈ 𝐶) → (A ↾ ran B) ∈ V)
13		rnexg 4540	. . . . 5 ⊢ ((A ↾ ran B) ∈ V → ran (A ↾ ran B) ∈ V)
14	12, 13	syl 14	. . . 4 ⊢ ((Fun A ∧ B ∈ 𝐶) → ran (A ↾ ran B) ∈ V)
15	9, 14	syl5eqel 2121	. . 3 ⊢ ((Fun A ∧ B ∈ 𝐶) → ran (A ∘ B) ∈ V)
16		xpexg 4395	. . 3 ⊢ ((dom (A ∘ B) ∈ V ∧ ran (A ∘ B) ∈ V) → (dom (A ∘ B) × ran (A ∘ B)) ∈ V)
17	8, 15, 16	syl2anc 391	. 2 ⊢ ((Fun A ∧ B ∈ 𝐶) → (dom (A ∘ B) × ran (A ∘ B)) ∈ V)
18		ssexg 3887	. 2 ⊢ (((A ∘ B) ⊆ (dom (A ∘ B) × ran (A ∘ B)) ∧ (dom (A ∘ B) × ran (A ∘ B)) ∈ V) → (A ∘ B) ∈ V)
19	3, 17, 18	sylancr 393	1 ⊢ ((Fun A ∧ B ∈ 𝐶) → (A ∘ B) ∈ V)

Syntax hints:   → wi 4   ∧ wa 97   ∈ wcel 1390  Vcvv 2551   ⊆ wss 2911   × cxp 4286  dom cdm 4288  ran crn 4289   ↾ cres 4290   ∘ ccom 4292  Rel wrel 4293  Fun wfun 4839

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853

This theorem is referenced by:  cofunex2g  5681